.( 1.. )
i i
W Q q i n
δ = δ =
A generalized force Qj contributes to δW only if the corresponding
generalized coordinate qj is given a virtual displacement. ( independent !)
: Virtual work δW of the actual forces for each individual variation of only the generalized coordinates at a time.
Since the transformations are invertible, a single variation δq j will induce a simultaneous variation of one or more of the physical coordinates.
A virtual displacement of a generalized coordinate in physical space ~ A combination of virtual displacements subjected to the constraints of
the system.
Generally, the corresponding virtual work done by the physical components of
the forces can be computed and set equal to Q(i)δ q(i). Example: Figs.2.9, 2.10
Consider a spring-loaded cart ( Fig.2.4 ) with a swinging pendulum attached to it.
This system has two degrees of freedom. Chosen x and θ as generalized coordinates.
Since x and θare independent variables,
x x
δ = δ
,δθ = 0
(2.34)And δ =x 0 , δθ = δθ (2.35)
are two sets of admissible virtual displacements.
Now compute the corresponding virtual work done by the external forces under each of the designated virtual displacements :
If δ ≠x 0 and δθ = 0
W F xs
δ = − δ (2.36) If δ =x 0and δθ ≠ 0 (Fig. 2.10),
sin
W mgl θ θ
δ = − δ (2.38)
sin
Qθ mgl θ
∴ = − (torque) (2.39)
Then for an arbitrary combination of virtual displacements, the total virtual work is
sin
W kx x mgl θ θ
δ = − δ − δ (2.40) Note :
Physical interpretation of a generalized force depends on the significance of the related generalized coordinate.
Once a given set of generalized coordinates are specified, the generalized forces can in principle always be determined, regardless of the physical interpretation of the generalized coordinates.
~ Holonomic systems, the computation of generalized forces is very simple. : Virtual work done by holonomic constraint forces under a set of arbitrary virtual displacements compatible with the constraints is equal to zero.
Therefore, in the computation of generalized forces, only the applied forces need to be considered.
This results in a considerable benefit in the formulation of the equations of motion in terms of the generalized coordinates.
Special consideration may be given to conservative forces. Suppose that all the
forces acting on a system of N particles are conservative. Each physical force is derivable from a potential function.
Suppose: a single potential function:
1 1 1 2 2 2 ,
( , , , , , , , N, N N )
V =V x y z x y z x y z (2.41) The force on the i-th particle may be obtained as
i i
F = −∇V (2.42)
Where the gradient ∇i denote the operator
ˆi+ ˆj kˆ
i
i i i
x y z
∂ ∂ ∂
∇ = +
∂ ∂ ∂
Substituting the physical components of the forces (2.42) into Equation (2.31) results in the characterization of the virtual work as the negative of the variation of the potential function :
1
( )
N
i i i
i i i i
V V V
W x y z V
x y z
=
∂ ∂ ∂
δ = − δ + δ δ = −δ
∂ ∂ ∂
∑ (2.44)
Thus the virtual work done by a collection of conservative forces, under specified virtual displacements, is given as the negative of the variation of potential energy.
Principle of Virtual Work:
A conservative system is in static equilibrium iff the total potential energy of the system is stationary
0
δ =V (2.45) Thus the virtual work done by a collection of conservative
forces, under specified virtual
displacements, is given as the negative of the variation of
potential energy.
Principle of Virtual Work:
A conservative system is in static equilibrium iff the total potential energy of the system is stationary
0
δ =V (2.45) Suppose : Single potential function
1 1 1 2 2 2 ,
( , , , , , , , N, N N )
V =V x y z x y z x y z
Force on i-th particle may be obtained as
i i
F = −∇V (2.42) in here
ˆi+ ˆj kˆ
i
i i i
x y z
∂ ∂ ∂
∇ = +
∂ ∂ ∂ ≡
Substitute Eqn.(2.42) into δW : Eqn.(2.31) ,
1
( )
N
i i i
i i i i
V V V
W x y z V
x y z
=
∂ ∂ ∂
δ = − δ + δ δ = −δ
∂ ∂ ∂
∑ (2.44)
: Virtual work done by conservative forces ~
Negative of the variation of potential energy.
Principle of Virtual Work: (MINI ? MAX ?)
0
δ =V Principle of Stationary Potential Energy ~
Necessary and sufficient condition for static equilibrium
of a conservative system.
Transforming to generalized coordinates : Total P.E. of a conservative system as
1 2
( , , , n)
V =V q q q (2.46) Consequently, the variation of the P.E. function in terms
of δqj is : Eqn(2.47)
or
1
, j
n
j q j
j j
W V q V q
= q
δ = − ∂ δ ≡ − δ
∑ ∂ ( for j=1..n) (2.48) For a conservative system :
Generalized forces ~ also derivable from a potential function in terms of the generalized coordinates qj . That is,
j
j
Q V
q
= − ∂
∂ (2.49) Therefore the determination of generalized forces for
conservative systems is very easy (?) Using transformation of coordinates
~ As a final step in the derivation of equations of motion
After the change of variables has been consummated, we will only need to keep the final result.
LAGRANGE’S EQUATIONS OF MOTION
Up to now, we consider the connection between physical variable and generalized coordinates based on the
geometric configuration of a system ( admissible !).
Especially, generalized coordinates compatible with the constraints make the kinematics much more manageable for holonomic systems
We are now in a position to make the transition between vector mechanics and analytical mechanics.
Instead of using free - body diagrams :
Based on the variation of energy and the minimum number of coordinates needed to characterize the
dynamics of the system ( always possible ?).
: Lagrangian dynamics !!
Kinetic energy, potential energy, and virtual work are all
scalar quantities. Thus, the transformation of these
quantities is rather straightforward.
Based on a system qj instead of the physical coordinates ri. - A unified approach in a way that is independent of any
particular coordinate system or set of generalized coordinates.
For a system of N particles subjected to only holonomic constraints. The more general case will be considered later.
Assume a system with n degrees of freedom and that
there is a transformation :
For the i th particle in a vector form as
mi ai = Fi (2.50) or
pi
i
d F
dt = (2.51) : linear momentum of the i-th particle as
pi = mi ir (2.52) Find out how the equations of motion transform under
the transformation to generalized coordinates.
d (..)
dt ? :
Generalized momentum corresponding to the k th generalized coordinate is given by
( ) ( )
k k
k
d d T
p p
dt dt q
= = ∂
∂
(2.53) By definition, the total kinetic energy of the system is
2 2 2
1
1 ( )
2
N
i i i i
i
T m x y z
=
=
∑
+ + (2.54)then the generalized momentum pk as
1
( )
N
i i i
k i i i i
k i k k k
x y z
p T m x y z
q = q q q
∂ ∂ ∂
= ∂ = + +
∂
∑
∂ ∂ ∂ (2.55) Remember the chain rule :
1 N
i i
i j
j j
x x
x q
q t
=
∂ ∂
= +
∂ ∂
∑
Then take derivative wrt qk :
i i
k k
x x q q
∂ ∂
∂ = ∂
(2.57) Thus, each component pk can be expressed as Eqn.(2.58)
Taking the total time derivative of Eqn (2.58) and
applying the product rule to the terms in the summation ( Remember : dtd (x y) x y x y
• • • ••
= + )
1
( ) ( )
N
i i i
i i i i
k i k k k
x y z
d T
m x y z
dt q = q q q
∂ ∂ ∂
∂ = + +
∂ ∑ ∂ ∂ ∂
1
[ ( ) ( ) ( )]
N
i i i
i i i i
i k k k
x y z
d d d
m x y z
dt q dt q dt q
=
∂ ∂ ∂
+ + +
∂ ∂ ∂
∑
Remind the terms in the first summation as the Newton’s Second Law
i i ix
m x = F m yii = Fiy m zi i = Fiz
Thus the terms can be rewritten as
1 1
( ) ( )
N N
i i i i i i
i i i i ix iy iz
i k k k i k k k
x y z x y z
m x y z F F F
q q q q q q
= =
∂ ∂ ∂ ∂ ∂ ∂
+ + = + +
∂ ∂ ∂ ∂ ∂ ∂
∑ ∑
where the right-hand side ~ generalized force Qk given by the transformation equations.
To interpret the second summation terms in Eqn(2.59), note that
2 2
1
( )
n
i i i
j
k j j k k
x x x
d q
dt q = q q t q
∂ = ∂ + ∂
∂ ∑ ∂ ∂ ∂ ∂
[ jn1 i j i ] i i k,
k j k
x x
q x x
q q t q
• •
=
∂ ∂
∂ ∂
= ∂ ∑ ∂ + ∂ = ∂ ≡
Thus the time rate of change of the kth generalized momentum is given by Eqn(2.60)
Finally, the equations of motion in terms of qk :
( ) k
k k
d T T
dt q q Q
∂ ∂
− =
∂ ∂ k = 1, 2,, n (2.61)
: General form of Lagrange’s Equations of Motion
There is one equation corresponding to each qk.
The system of equations represents a coupled system of
